Obliquity: A Probe for the Spin-Orbit Alignment and Formation History of Hot Jupiters

A Thesis Presented by Caleb Cañas Submitted to

The Department of Astronomy in partial fulfilment of the requirements for the degree of Bachelor of Arts

Thesis Advisor: David Latham

April 11, 2014 – 2 –

Obliquity: A Probe for the Spin-Orbit Alignment and Formation History of Hot Jupiters

Caleb Cañas

ABSTRACT

Hot Jupiters are intriguing because of their close proximity to the host star. Current theories have tried to explain their formation. However, these theories require knowledge of the obliquity, or the projected angle between the stellar spin vector and the orbital spin vector, for validation. In this thesis, I explore what the obliquity can inform us about the alignment between the planetary orbit and host star. I analyze the Rossiter-McLaughlin effect to derive the obliquity for two Hot Jupiter systems: HAT-P-49 and KELT-7. I use photometric data from KeplerCam to estimate stellar and systemic parameters. I use spectra from TRES and SOPHIE to estimate radial velocities and derive orbital solutions. For these systems, I adopt a simple model for the Rossiter-McLaughlin signature derived from geometry. I constrain the obliquity with this model using the MCMC analysis method. HAT-P-49b is a 1:7 MJ planet orbiting a 6820 K 1:5 M ˇ

F-type star with a calculated obliquity of  90 17ı. KELT-7b is a hot  ˙ Jupiter with a calculated obliquity of  8 17ı orbiting a 6780 K F-type star.  ˙ I explore the possibility of constraining the obliquity for these systems using the rotational period, the stellar radius, and the projected rotational velocity of the host star. The potential for this second method is limited by the accuracy of the stellar parameters used. – 3 –

Acknowledgements

This thesis would not have been possible without the support and collaboration of many people. I would like to thank David Latham for his patience, guidance, and time during the past year. This was my first time researching in astrophysics and I am grateful forbeing introduced into the exciting world of exoplanets. Joshua Winn was another invaluable source for this project. His knowledge and guidance in all things Rossiter-McLaughlin helped make the analysis of HAT-P-49 and KELT-7b possible. I would like to thank Allyson Bieryla for her support, encouragement, and assistance with data management. I thank Lars Buchhave for providing the rotational velocities of KOI host stars and his knowledge on the derivation of the rotational velocities for stars. I thank the observers at Fred Lawrence Whipple Observatory who collected the data I used in this thesis.

I would like to thank Jim Moran for giving me an opportunity to write a senior thesis despite being from a different concentration and for his insight, support and encouragement throughout year. I am grateful to my peers in Astronomy 99: Adrian Arteaga, Brian Claus, Diana Powell, and Natania Wolansky for their support and for making the year truly enjoyable. A shout out goes to my roommate, Chi Zeng, for his willingness to listen as work on my thesis progressed and for his optimism. Finally, I am grateful to friends and family, especially my mother, for their continuous support throughout my thesis and in everyday life. – 4 –

Contents

1 The Importance of Extrasolar planets 7

1.1 A brief history on the quest for exoplanets ...... 8

1.2 Hot Jupiters: Enigmas in planetary formation ...... 9

1.2.1 Purported Migration of hot Jupiters ...... 10

1.3 Measuring Stellar obliquity ...... 17

1.3.1 The Rossiter-McLaughlin Effect ...... 17

1.3.2 Spectroscopy, Photometry, and Obliquity ...... 22

1.3.3 Asteroseismology and the Constraint on the Obliquity ...... 27

2 Observations 31

2.1 HAT-P-49b ...... 31

2.2 KELT-7b ...... 32

3 Obliquity from the Rossiter-McLaughlin Effect 33

3.1 HAT-P-49b: A Polar Hot Jupiter ...... 33

3.1.1 Validating the Residual Velocity of HAT-P-49b ...... 33

3.1.2 Modeling the RM Effect in HAT-P-49b ...... 34

3.1.3 Results and Discussion for HAT-P-49b ...... 40

3.2 KELT-7b: An Aligned Hot Jupiter ...... 41

3.2.1 Determining the Transit Parameters for KELT-7b ...... 41 – 5 –

3.2.2 Modeling the RM Effect in KELT-7b ...... 47

3.2.3 Results and Discussion for KELT-7b ...... 47

3.3 Improved Modeling of the RM Effect ...... 48

4 Constraining the Obliquity with Stellar & Transit Parameters 55

4.1 Required Parameters ...... 55

4.1.1 Stellar Parameters ...... 55

4.1.2 Spectroscopic Rotational Velocities ...... 56

4.1.3 Stellar Rotational Period ...... 57

4.2 Results ...... 58

4.2.1 Potential Error in the Stellar Radius ...... 58

4.2.2 Potential Error in the Rotational Period ...... 63

4.2.3 Potential Error in the Rotational Velocity ...... 65

5 Conclusion 66

5.1 Obliquity and the RM Effect ...... 66

5.2 Obliquity and Stellar Parameters ...... 68

5.3 The Obliquity and Exoplanetary Formation ...... 69

A The Radial Velocity Solution 71

A.1 Orbital Elements ...... 71

A.2 Deriving the Radial Velocity Solution ...... 73 – 6 –

B Derivation of the Rossiter-McLaughlin Effect 74

B.1 The Geometry of a Transiting Extrasolar Planet ...... 74

B.2 Parameters for the system ...... 75

B.3 Analytical Approximation ...... 78

B.4 Addition Offset Velocity for HAT-P-49b & KELT-7b ...... 83

B.5 The semi-amplitude velocity as a free parameter for the RM Effect in HAT- P-49b & KELT-7b ...... 83

C Correlation Plots 85

C.1 HAT-P-49b ...... 85

C.2 KELT-7b ...... 85

D Markov Chain Monte Carlo Procedure for Uncertainties 85

References 89 – 7 –

1. The Importance of Extrasolar planets

One of the questions humanity has always grappled with is about intelligent life elsewhere in the Universe. Is there another Earth somewhere in space? Throughout the centuries, particularly after the era of Greek philosophers, speculation about other worlds emerged and ranged from Aristotle’s assertion that “here cannot be more worlds than one” to Epicurius’ optimistic notion that “there are infinite worlds both like and unlike this world of ours”. Until recently it was impossible to confirm or refute any of these arguments. Fora long time, the only planetary system for study was the Solar System, from which subsequent theories about planetary formation and dynamics were derived. Fortunately, the recent discovery of extrasolar planets, or exoplanets, has been a boon to astronomers interested in planetary systems, their formation, their processes, habitability and, ultimately, about life elsewhere.

Beginning in the early nineties, exoplanets were discovered using a variety of methods including the transit method, radial velocity method, astrometric measurement, direct imaging, and gravitational microlensing. As of this thesis, the NASA Exoplanet Archive lists 3845 Kepler candidates of interest and 1696 confirmed exoplanets. Of the 1696 confirmed exoplanets, most have been detected via the radial velocity method (503 exoplanets) or the transit method (1116 exoplanets). The radial velocity method was used to detect the first and exoplanet and looks for the host star’s reflex orbit due to the presence of a planet. The reflex orbit manifests itself as a perturbation in the radial velocity of the star.The transit method looks for a decrease in the light received from a system as a planet crosses in front of the disc of its host star. There are numerous transiting exoplanet surveys such as WASP, XO, CoRoT, TrES, and the well-known Kepler mission. All of these surveys aim to identify exoplanets and to improve models of planetary formation and dynamics. The goal of NASA’s Kepler mission was to observe transiting exoplanets in the search for Earth- – 8 – like candidates to give astronomers an idea of their frequency within the Milky Way. The Kepler mission has discovered a total of 964 confirmed exoplanets (from the NASA Exoplanet Archive). Astronomers have discovered exoplanetary systems with varying configurations and properties ranging from highly eccentric hot gaseous giants, exoplanets with turbulent evaporating atmospheres (Hébrard et al. 2004), and some with spectral lines suggesting a watery atmosphere (Swain et al. 2009; Deming et al. 2013). Recently, with the Kepler data, it has been speculated the Milky Way should host roughly forty billion planets with planets with the nearest Earth-like planet about four parsecs away (Petigura et al. 2013).

1.1. A brief history on the quest for exoplanets

New instrumentation and techniques have revolutionized planetary discovery in the past decade. Before the nineties, the existence of exoplanets was purely theoretical. Long before the term exoplanet was in use, van de Kamp (1963) believed he discovered a companion to Barnard’s star. He used roughly fifty years of observations to build a case for an alleged

companion with a mass of 1:6 MJ. At the time, this was such a novel concept that the New York Times declared there was another Solar System in our Universe. Years of scrutiny and observations failed to confirm the purported discovery and, recently, Choi et al. (2013) gave substantial evidence against the possibility of any planet around Barnard’s star and instead suggested it was purely geometric effect.

Two of the earliest observations searching for exoplanets were performed separately by Bruce Campbell (Campbell et al. 1988) and by David Latham (Latham et al. 1989). While at the time these initial observations were not sufficient to prove the existence of exoplanets, they mark the start of a boom in exoplanetary discoveries. Campbell et al. observed slight perturbations in the radial velocity residuals, but dismissed it as activity in the Ca II emission line with a period comparable to those of the perturbations (Walker et al. 1992). Only later – 9 – would it be shown that this was a hot, gaseous giant now called γ-Cephi (Hatzes et al. 2003). Latham et al. similarly observed periodic variations in the radial velocity of the star HD114762 and hypothesized the unseen companion was either a planet or a brown dwarf. A decade later, evidence would suggest that this was a planetary candidate (Marcy et al. 1999), but as of now there is difficulty in confirming non-transiting exoplanets. The first conclusive evidence of an exoplanet was that of the multi-planetary system around the pulsar PSR B1257+12 in 1992 (Wolszczan & Frail 1992). While the highly radiative environment impeded these planets from hosting organic life as we know it, for the first time there was evidence that planets were not unique to the Solar System.

The next major discovery intrigued many astronomers. In 1995 51 Pegasi b, the first exoplanet around a main sequence star, was observed (Mayor & Queloz 1995; Marcy & Butler 1995). Astronomers were shocked when the calculations implied it had a period of roughly four days, suggesting a massive Jupiter-like planet at a distance of roughly 0:05 AU from its host star (for reference, Mercury has a mean semi-major axis of 0:39 AU). At the time, 51 Pegasi b was incompatible with the theories of planetary formation solely derived from our Solar System. Astronomers now know it is the prototype for a class of planets dubbed hot Jupiters. Hot Jupiter is a term referring to giant gaseous planets orbiting extremely close to the star with periods less than ten days and often with eccentric orbits. Following this discovery, various other hot Jupiters have been discovered and analyzed through expansive exoplanetary surveys.

1.2. Hot Jupiters: Enigmas in planetary formation

Despite all of these discoveries, we lack have a complete model for planetary formation and evolution. The properties and characteristics of newly discovered systems are more complex than current theories predict. A robust theory for planetary formation is one – 10 – which could describe the Solar System, exoplanets masses and frequencies, the propensity of exoplanets to have high eccentricity, and their distances from the star. For a hot Jupiter there is currently no possible method of formation near its star. I will focus on what observations can tell us about hot Jupiter formation. Models based on our own planetary system suggest gaseous giants form beyond ice line, the point where water freezes. This model, called core accretion, is built on the assumption that a seed planet, or core, grows via collisions in a protoplanetary disk (Armitage 2010). The core must reach a critical mass before the gaseous disk dissipates. If this condition is satisfied, a hydrodynamic instability results in the rapid accretion of gas onto the core. The critical mass for a core is on the order of 10 M and the ˚ end result is a gaseous planet resembling Jupiter or Saturn. The presence of ice provides the mass necessary for a large seed planet and for core accretion to create such enormous planets. The rate of growth for the core is controlled by the surface density of rocky and icy bodies in the disk and by the amount of gravitational focusing (Armitage 2010). Inside the ice line, where water is not in solid form, there is not enough mass to form giant gaseous planets. In an attempt to correct this discrepancy, astrophysicists hypothesize hot Jupiters form outside of the ice line and either migrate or are flung inwards until they become stabilized inthe configuration we observe.

1.2.1. Purported Migration of hot Jupiters

The theory of inward migration brings to question which mechanisms could allow a hot Jupiter to move from its relatively cold cradle to the hot stellar neighborhood. There are three main theories for planetary migration: migration through interaction with a gaseous protoplanetary disc, interactions between planets and remnant planetesimals, and interactions between two or more separate planets. Even before the first hot Jupiter was observed, it was hypothesized that angular momentum could be transferred from a planet – 11 – to its progenitor disc, leading to inward migration of the planet (Goldreich & Tremaine 1980; Lin & Papaloizou 1986a,b; Ward & Hourigan 1989). For hot Jupiters, this gas disc migration is known as the Type II regime (Lubow & Ida 2011). A massive planet can distort the gaseous disc (Figure 1.1) by creating an annular gap. This reduces the gravitational torque on the planet and causes migration at the inflow rate of gas.

Planetesimal-planet interactions can cause an inward migration due to the recoil from gravitational interactions with planetesimals (Ormel et al. 2012). These planetesimals can approach from the interior or exterior of the planetary orbit. The gravitational interaction causes the planet to undergo a random walk and, if the scattering is large enough, an inward migration occurs (Figure 1.2). Finally, if the initial planetary system is unstable, planet- planet interactions can cause ejection of one planet and lead to inward migration of the other planet (Levison et al. 1998). The planet which has migrated inward is placed in an eccentric orbit which circularizes due to tidal interactions with the star as time passes. Another possible migration mechanism is one where the orbit of the planet shrinks by the combined effects of secular perturbations from a distant companion star, known as Kozai oscillations, and tidal friction (Fabrycky & Tremaine 2007). This mechanism is most likely in a binary system and is probably not the major source for hot Jupiter migration.

Gas disc migration and planetesimal disc migration are hypothesized to cause small perturbations in the orbital motion. During growth, the protoplanet has tidal interactions with the gaseous disk. A gaseous giant will carve a gap in the disk such that growth of the planet halts. However, once the gap is formed the transfer of angular momentum continues. The protoplanet will therefore undergo orbital migration which is coupled to the viscous evolution of the disk (Lin et al. 1996). The planet migrates inward retaining the angular momentum vector of the disk. Assuming the rotation of the star is well aligned with the disc, it will also be well aligned with the planet. Dynamical processes, such as planet-planet – 12 –

Fig. 1.1.— Modes of Planetary-Gas Migration The different types of planetary migration are shown above. In Type I migration, alow mass planet excites a density wavefunction in the disc. The planet follows this gas flow into the disc. Type II migration is thought to occur with massive exoplanets. A massive planet can carve a gap in the disc, effectively interrupting accretion. If the disc is strong enough we see accretion continues and the planet migrates inward. Type III migration is a special case where corotational resonance modifies the type I migration rate, resulting in a rapidly migrating gap with a velocity dependence on the mass accumulation rate. This is thought to be a rare occurrence and not a primary method of migration. The images for type I and type II migration are from Armitage & Rice (2005). The image for type III migration is adapted from Frederic Masset (Source). interactions and Kozai migration, are hypothesized to create very misaligned hot Jupiters due to the randomness of the interactions. In order to distinguish among the possible theories, an understanding of the spin orbit alignment is necessary. The stellar obliquity is shown in Figure 1.3. It is the angle between the spin vector of the star and the spin vector of the planetary orbit, can shed light on the orientation of the planets orbit with respect to the stars rotation and eventually identify which of these theories is most consistent with all of the planetary observations.

Measurement of the obliquities of hot Jupiters has revealed a diverse group of planets, some which are highly misaligned and others which are coplanar with the host star (Fabrycky & Winn 2009; Winn et al. 2011). This has led to the hypothesis that after migration hot Jupiters undergo tidal evolution Winn et al. (2010). A hot Jupiter migrates to the environment around the star through some dynamical processes (Albrecht et al. 2012). Due – 13 –

Fig. 1.2.— Migration through body interactions I. A sample of planet-planet interaction migration. The current configuration of our Solar System is hypothesized to be a result from the migration of the gas giants. This graph shows the initial semi-major axis of Jupiter, Saturn, Uranus, and Neptune and their subsequent evolution through time via simulations using the Nice model. The three curves for each planet indicate the minimum heliocentric distance, q, the semi-major axis, a, and the maximum heliocentric distance Q. The image is from (Tsiganis et al. 2005). II. The numerical simulations of the lunar heavy bombardment (LHB) depicts what may happen for a dynamically unstable system. The four points are at (a) the start of migration, (b) prior to the start of the LHB, (c) after start of the LHB, (d) 200 Myr after the LHB. The orbits of the four gas giants alter with time. The image is from Gomes et al. (2005). – 14 –

Fig. 1.3.— The Projected Spin-Orbit Angle This is view of a transiting exoplanet the observer sees. The actual angle between the vectors cannot be determined from the observer’s perspective, but the apparent angle between the spin vectors can. The spin vector of the planetary orbit is p and the rotational spin vector of the host star is  . The obliquity is denoted as  in the figure. In the above configuration,  the Z-axis points towards the observer and the Y -axis is taken to be the stellar spin vector. For the observer, the sky is the XY plane. The coordinate system used for calculations is a simple rotation about the X-axis by .90 i/ı. – 15 – to the nature of the migration, a wide range of values for the spin-orbit alignment is expected. A planet near a cool star is expected to come into alignment rather quickly (on order a few Gyrs) due to the presence of a thick convective envelop around the cool star to facilitate tidal dissipation. A planet near a hot star is not realigned in short timescales because the host lacks a thick convective envelope. From observations, it was found that hot Jupiters with high obliquities tend to be located near stars with Teff > 6520 K (Figure 1.4) while hot Jupiters with nearly aligned orbits to the stellar rotation tend to be found around cooler stars (Winn et al. 2010). One interpretation of these results is that hot Jupiters form primarily from the interaction and scattering with other celestial bodies. Misaligned exoplanets would have migrated via some chaotic and dynamic process. They should have migrated recently because tidal interactions with the host star have not realigned these planets.

The theory proposed by Albrecht et al. (2012) assumes the stellar equatorial plane was originally aligned with the protoplanetary disk. However, this does not necessarily have to be the case and there are other theories in hot Jupiter migration. Other models for the observed misalignments include chaotic stellar cradles (Bate et al. 2010), the capture of gas from neighboring stellar companions (Thies et al. 2011), magnetic interactions between a young star and the accretion disc (Foucart & Lai 2011; Lai et al. 2011), random directional wandering as a result of the transport of angular momentum by internal gravity waves (Rogers et al. 2012), and gravitational torques from massive and distant bodies as the protoplanetary disc undergoes dynamical evolution (Batygin 2012). In the previous theories, hot Jupiter migrate through gaseous interactions but the reason for any significant misalignment is due to the perturbation on the star. These theories suggest that large obliquities should be present in not just hot Jupiter but in all exoplanets. The only way to test any of these theoretical models for inclination is to measure the obliquity and find which model is consistent with the observed values. – 16 –

Fig. 1.4.— Distribution of stellar spin-orbit angles The projected obliquity is plotted as a function of the stellar effective temperature. Red stars are those with temperatures high than 6250 K while blue stars are those with temperatures less than 6250 K. Black dots are systems with periods larger than 7 days and host planets of mass < 0:2 MJ. Note the trend that cool stars have very low obliquity while hotter stars have higher obliquity and by extension more misalignment between the spin vector of the orbital plane and the spin vector of the star. The figure is from Albrecht et al. (2012). – 17 –

1.3. Measuring Stellar obliquity

The test of theories above rely heavily on the measurement of the obliquity. As of this thesis, there have been roughly 79 systems for which the spin-orbit alignment has been measured using the Rossiter-McLaughlin effectŽ. Other techniques include the analysis of stroboscopic star spots (Sanchis-Ojeda & Winn 2011), stellar rotational statistics (Schlaufman 2010), gravitational darkening of light curves from rapidly rotating stars (Barnes 2009; Barnes et al. 2011), analysis on rotational periods and velocities (Hirano et al. 2012b), and long-term variations in the transit duration (Damiani & Lanza 2011). I will focus on the Rossiter-McLaughlin effect and using rotational velocities and periods to derive the obliquity. I will also discuss the use of asteroseismology, a recent technique with promise in deriving obliquities of exoplanets (Chaplin et al. 2013).

1.3.1. The Rossiter-McLaughlin Effect

In 1924, the McLaughlin and Rossiter each published papers in the Astrophysical Journal describing the photometric and spectroscopic observations (McLaughlin 1924; Rossiter 1924) which could be analyzed to estimate the projected angle of obliquity. The phenomena was originally conceived of as a rotational effect in eclipsing binaries. McLaughlin used the method to give the alignment in the Algol system while Rossiter produced the alignment for the β Lyrae system. The apparent velocity deviation due to the companion crossing the stellar disc, later named in honor of them, was calculated by McLaughlin and

1 1 Rossiter to be 35 [km s ] and 26 [km s ], respectively. In 2000 it was hypothesized that the Rossiter-McLaughlin effect could help confirm an exoplanetary transit, determine the transit

ŽI reference René Heller’s Holt-Rossiter-McLaughlin Encyclopaedia, a compilation of the spin-orbit alignment measurements from the literature. Source: http://www.physics.mcmaster.ca/~rheller – 18 – latitude, and constrain the orientation between the stellar spin axis and the orbital plane (Schneider 2000). The first successful measurement of the Rossiter-McLaughlin (RM) effect came in the same year (Figure 1.5) and it was on the system HD209458 (Queloz et al. 2000). The exoplanet was found to be well-aligned with the star. Since this initial observation, numerous systems have been analyzed and characterized using the effect (Dreizler et al. 2009; Fabrycky & Winn 2009; Triaud et al. 2010). −

Fig. 1.5.— The first exoplanetary Rossiter-McLaughlin signal I. The first Rossiter-McLaughlin effect of an exoplanet was measured by Queloz andMayor in 2000. The planet was calculated to have a spin orbit angle of  3:9ı. The red hatched  pattern indicates the duration of the transit and it is in agreement with the distortion in the residual velocity. II. The top graph is the measurement performed by McLaughlin of the Algol system while the bottom is the measurement performed by Rossiter on the β-Lyrae system. While the scales are completely different, the spectroscopic phenomena observed by Queloz and that of Rossiter or McLaughlin share the same physical principle and therefore produce a blip in the radial velocity curve.

The RM effect (Figure 1.6) manifests itself when the transiting body casts a shadow – 19 – on the star. The rotation of the star will cause an additional redshift or blue shift over the Doppler shift from its bulk motion through space. Typically, half of the stellar light should be redshifted while the other half should be blueshifted. Consequently, the spectroscopic lines will be distorted, if a planet casts a shadow on our stellar surface. The planet obstructs the additive Doppler shift from the spin and results in an observable signature on the line profile of the star. If the stellar spin is the dominant cause of line broadening, thiswill directly translate to an observable distortion of the radial velocity curve. The radial velocity of the star is measured using these line profiles and a perturbation in the velocity produces residuals that give the characteristic blip of the RM effect (Figure 1.7).

Fitting the RM effect can give an estimate of the stellar obliquity. For simplicity, onecan assume normalized Cartesian coordinates centered on the star. In this coordinate system, the amplitude of the RM effect is only dependent on the velocity as a function of its position across the stellar disc, X. With knowledge of a few other parameters the effect can be roughly modeled as:

vRM V sin i X .t; / ı    X .t; / normalized position (1.1) D ı fractional loss of light D The derivation of this expression is in the appendix. In the above expression, the amplitude

is given as KRM V sin i while the shape is defined by a dimensionless function g .t; / D   D X .t; / ı. The value of ı is determined by the limb darkening law assumed. If one completely 2 R ignored limb darkening, the fractional loss of light is simply ı p . Typically, to have D Rs this a robust parameterized form, it is important to take into account  limb darkening. One can constrain the value of  by fitting the observed residuals.

The approximation is an oversimplification of the actual system. I treat the velocity as one dimensional and perpendicular to my line of sight, I ignore the effects of turbulence within the stellar atmosphere, and, most importantly, the fact that I am treating a spectral – 20 –

Fig. 1.6.— The Rossiter-McLaughlin effect: a spectroscopic phenomenon The above figure describes the physics of the Rossiter-McLaughlin effect. The firstrow shows the shadow on the stellar surface cast by the planet as it transits, the second row shows the surface radial velocity. The third row shows the line profile an observer records if line broadening is dominated by stellar spin, in which case the perturbation caused by the transiting planet is noticeable. The figure is adapted from Gaudi & Winn (2007). – 21 –

λ − − −

Fig. 1.7.— The RM distortion of the radial velocity The above plots are three different simulations of the Rossiter-McLaughlin effect forthe given impact parameter and projected spin-orbit angle. All other parameters are the fixed. Note the RM signal varies with angle, making it possible to distinguish the orientation of the spin vector of the planetary orbit with respect to the stellar spin axis. The blue solid line is a calculation with a linear limb darkening law while the red line neglects limb darkening. Image adapted from Gaudi & Winn (2007). – 22 – perturbation as a pure Doppler effect. The computations of angles done with Equation 1.1 do not depend on the distorted shape of the line. It gives the first moment, or the intensity- weighted mean wavelength, of the distortion. However, this would certainly deviate from a result with more realistic conditions and approximations. More robust techniques for fitting the RM effect involve cross-correlation of simulated spectra with actual spectrato model the perturbation as the difference between the maximum of the correlation function. While one way to address this issue is to simply model the spectral distortion of the lines (Collier Cameron et al. 2010), this is usually not an ideal situation. An analytical expression has been developed (Hirano et al. 2011) where the intrinsic broadening, rotational broadening, and micro/macroturbulence of the system has been incorporated to yield an equation for the RM distortion (Figure 1.8) that can be applied to any system. An analytical solution is less computationally expensive than the cross-correlation method and the work by Hirano et al. has improved both our understanding of the RM effect and our ability to model it.

1.3.2. Spectroscopy, Photometry, and Obliquity

Hirano et al. recently (Hirano et al. 2012a,b) used spectroscopy and photometry to derive the obliquity for some Kepler planetary candidates. The RM effect is most readily detected in bright, rapidly rotating stars above the twelfth magnitude and for larger Jupiter- sized planets. While the RM effect has yielded a calculation for super-Neptunes (Winn et al. 2010), smaller planets and fainter stars could potentially be masked from detection. The use of stellar parameters in deriving the obliquity does not depend on the amount of distortion the planet can cause to the spectral lines. Instead, if one could determine the spectroscopically

derived projected equatorial rotational velocity, .V sin i /spec, the rotational period of the   – 23 –

Fig. 1.8.— Comparison of analytical models for the RM effect Hirano et al. (2011) developed an analytical solution to the RM effect. The plot above depicts the analytical solution for the RM effect in HAT-P-4. Equation 16 refers to the equation derived in the paper, Gaussian formula refers to the approximation Hirano made in 2010 where he included some turbulence, OTS refers to equations by Ohta et al. (2005) where only linear darkening is considered. The panel shows the residuals between the new analytical model and previous empirical models. Hirano et al. show the shape of the RM effect can ultimately be improved using parameters such as differential rotation and turbulence. – 24 – star P , and the radius, R , then the inclination from Hirano et al. (2012b) is given as: 

.V sin i /spec .V sin i /phot   D   2R V  (1.2)  D P P .V sin i /spec sin i    D 2R  The full three dimensional angles in a planetary system (Figure 1.9) can be related by using:

cos cos i cos ip sin i sin ip cos  D  C  Three dimensional angle D (1.3) i Stellar inclination  D

ip Orbital inclination D In order to actually observe transiting exoplanets, the orbital inclination must be very close

Fig. 1.9.— Three-dimensional angles of a planetary system The above figure shows the angles for a given planetary system. The observer canonly measure λ, which is the projected spin angle. The true obliquity is given as . To the left is a side-view of the system. The inclination is the angle between the stellar spin vector,  , and the observer. The project obliquity, , is the angle between the projection of    and the project of the exoplanet spin vector, p. depends on the stellar inclination, the projected obliquity, and the planetary inclination, ip. The right figure is how the system looks from the observer’s perspective.

to 90ı. Hirano showed that if one places limits on the stellar and planetary inclination, then – 25 – by considering from the three-dimensional angle and the transit parameters cos can be approximated: R 1 e sin $ 1 e sin $ cos ip b  C , since b C . 1 D a 1 e2 1 e2     R cos ip .  a (1.4)

cos cos i cos ip sin i sin ip cos  D  C  R cos .  cos i sin i , since sin ip cos  1 a  C  

In the approximation of cos iP , b is the impact parameter normalized to the stellar radius, e is the eccentricity, and $ is the argument of periastron. The approximation for

cos ip is only valid for transiting planets with almost circular orbits. The last line in Equation 1.4 gives a lower limit for the value of from the observed i . If the planet is almost aligned,  R R then i 90ı and cos a < 1. For most exoplanets, the value of a is small suggesting    there is a negligible value of for the system and that the spin vector of the star and spin vector of the planetary orbit should be fairly aligned. However, if i 90ı the system may   be misaligned because cos 0. That is to say, as the stellar inclination deviates from ¤ an angle of 90ı to the observer, the obliquity will increase for a transiting exoplanet. This method is limited based on the accuracy and precision of the stellar parameters. Hirano used the stellar parameters of few KOI to find those which may be misaligned (Figure 1.10). The systems where the stellar spin vector and the planetary orbit appear to be misaligned would have to be analyzed via the RM effect or with some other technique to get anactual value of . However, the use of stellar parameters does show promise in identifying potential systems with spin-orbit misalignment through an inclination difference.

Hirano used the radial period derived from observation of stellar spots. I intend to use the recent data for rotational period by McQuillan et al. where the period was determined by analysis of periodic photometric modulations from star spots. I will make use of the stellar radii derived by Huber et al. (2014) and radial velocities derived using cross correlation – 26 –

Fig. 1.10.— Constraints on obliquity from spectroscopy and photometry I. Plot of the effective stellar temperature and the stellar inclination using a rotational period estimated from the flux variation due to star spots, and rotational velocities and stellar radii derived from models using high resolution spectroscopy. The temperature distribution has significant correlations with what Albrecht et al. (2012) observed. A small value of is, or equivalently, a small value of sin is indicates misalignment. KOI numbers are listed next to each point. II. The plot shows the distribution of the equatorial velocity from sunspots and the projected equatorial velocity. The trends seems to imply that most of those exoplanets are slightly misaligned. Both figures are from Hirano et al. (2012b). – 27 – methods on stellar spectra.

1.3.3. Asteroseismology and the Constraint on the Obliquity

Asteroseismology seeks to extend measurement of obliquity to smaller Earth-like planets and even multiple planetary systems. Asteroseismic constraints of the obliquity are independent of planetary parameters, such as size and semi-major axis (Chaplin et al. 2013). Chaplin states that these methods of constraint require bright targets with high cadence, long-term observations. In the analysis, stellar parameters were derived from the stellar spectra and the frequency-power spectrum of the stellar light curve. The stellar inclination was derived using the broken degeneracy in the oscillation frequency for non-radial modes (Figure 1.11). Radial and non-radial oscillations in stars can be decomposed into spherical

m harmonics as a function of azimuthal and polar angles: Yl .; / (Aerts et al. 2010). For a spherical star, l is the degree while the azimuthal order, m, is degenerate. This degeneracy is broken by non-radial perturbations to the star, such as rotation. Non-radial modes are not spherically symmetric and any amplitude will be solely dependent on the viewing angle of the observer (Figure 12). With this theoretical model, Chaplin constrained the stellar inclination using MCMC. The following equation summarizes the calculations:

ı rotational frequency splitting  D 1 P D ı  (1.5) V sin i 2R ı sin i  D    P .V sin i / sin i   D 2R  As mentioned before, the degeneracy of oscillations is broken when the star rotates and this results in rotational splitting (Aerts et al. 2010). The rotational splitting, ı , is only  dependent on the azimuthal order m. Chaplin et al. (2013) assumed all of the rotational – 28 –

Fig. 1.11.— Asteroseismic constraint of obliquity These are theoretical profiles of a non-radial oscillation modeled with spherical harmonics, Y m, where the degree is l 1. The azimuthal angle m 1 is plotted in blue while the l D D m 0 component is red. Altering the angle of inclination alters the intensity of the individual D profiles. The rotational splitting is the difference between the red and blue lines andineach case is given as ı 1:5 μHz. The black profile is a combination of the two peaks. The  D panels in the left hand column correspond to the angles in Figure 1.12. The figure is from Chaplin et al. (2013). – 29 –

Fig. 1.12.— Intensity perturbations of the stellar surface with inclination The figure shows the intensity perturbations of the spherical harmonics for thedegree l D 1 with mode components at a pulse which corresponds to extreme displacement of the oscillations. The left column are patterns of azimuthal order m 1 while the right column D has azimuthal order m 0. The inclinations of the star with respect to the observer are D listed at in the middle. The point indicates the location of the spin vector and the dashed line is the stellar equator. Note how the change in inclination is accompanied by a change in the intensity of the stellar disk, allow observers to distinguish the orientation of the stellar spin vector to the observer. Figure from Chaplin et al. (2013). – 30 – splitting were equal which results in a variation of intensity with stellar inclination of the all terms in the spherical harmonics that are dependent on m. The rotational splitting was equated to the period of the star and in Equation 1.5, the stellar inclination is found by fitting the photometric data.

With this approximation of stellar inclination, we can revisit Equations 1.3 and 1.4 for systems much smaller than hot Jupiters. For the systems studied by Chaplin et al., the inclinations were found to be in agreement with fits from spectroscopy. Asteroseismology presents a novel approach to constraining the obliquity of a given system, especially that of smaller systems. Spin orbit angle have been measured for predominantly hot Jupiters but, if we seek to develop a theory of formation for terrestrial sized planets, it is necessary to probe their stellar alignment as well. – 31 –

2. Observations

2.1. HAT-P-49b

The planetary system HAT-P-49b was discovered and analyzed recently this year (Bieryla et al. 2014). As Bieryla et al. describes in the HAT-P-49b discovery paper, this is a hot Jupiter orbiting HD340099 discovered through the Hungarian-made Automated Telescope Network (HATNet). The planet was confirmed with photometric observations using the KeplerCam CCD system on the 1.2-meter telescope and spectroscopic observations using TRES (Tillinghast Reflector Echelle Spectrograph), the fiber-fed optical echelle spectrograph, on the 1.5-meter Tillinghast telescope both located at the Fred Lawrence Whipple Observatory (FLWO) in Arizona. Other spectroscopic observations were from the SOPHIE (Spectrographe pour l’Observation des Phénomènes des Intérieurs stellaires et des Exoplanètes) echelle spectrograph on the 1.93-meter telescope in the Observatory de Haute Provence (OHP).

Spectroscopic observations used in this thesis were collected with TRES. The date of observations range from June 2, 2012 to November 14, 2013. The transit event occurred on September 19, 2013. The photometric observations were obtained in fifteen-second exposures. Of the TRES data, the first five points had exposure times ranging from 180-450 seconds with a range in the signal to noise ratio of S 28 47. The derivation of the radial N  velocity measurements are described in detail by Bieryla et al.. Spectroscopic parameters were determined using Stellar Parameter Classification (SPC) (Buchhave et al. 2012). SPC is an IDL program which cross-correlates synthetic spectra created using Kurucz atmospheric models to the TRES spectra and estimates spectral parameters along with their uncertainties. Each spectrum was cross-correlated against the calculated template spectrum that gave the best correlation value, with a total of 14 spectral orders, for the velocity analysis. There were a total of fifty-three radial velocity measurements and each measurement had its respective – 32 – error estimated by the scatter in the velocities for the individual orders. The duration of the transit was approximately 4:1 hours. There were thirty-one radial velocity measurements outside of the transit while on the transit night there were: three measurements before ingress, sixteen measurements during transit, and three measurements after egress. To fit orbital parameters I used repack_multi_cc_generic, an IDL procedure by Lars Buchhave which implements a Levenberg-Marquardt least squares fitting algorithm. For the fit, I used all the data points except those inside the transit on the night September 19. That is to say, I fit the orbital solution without the purported RM effect so as to not remove saideffect.

2.2. KELT-7b

The spectroscopic data for KELT-7b were collected using TRES while the photometric data were collected with KeplerCam through a Sloan i-filter. The radial velocity measurements range from January 31, 2012 to October 19, 2013. There were a total of forty-eight radial velocity measurements, of which twenty-nine were on the transit night. The photometric event was observed with two-second exposures on the night of October 19, 2013 for a total duration of roughly 3.6 hours. There were a total of three separate light curves that we used to help derive the light curve parameters. The September 19, 2013 lightcurve was taken with a Sloan z-filter while the November 22, 2013 light curve was collected with a Sloan g-filter. Unlike HAT-P-49b, KELT-7b has yet to be published. I used the same IDL program by Lars Buchhave to derive the radial velocity solution. I then utilized the Transit Analysis Package (TAP) (Gazak et al. 2012) to fit the light curve and better constrain the orbital parameters such as limb darkening, semi-major axis, and transit depth. I did not adapt the photometric ephemeris for KELT-7b. For fitting the orbital parameters, I omitted data from the transit night. – 33 –

3. Obliquity from the Rossiter-McLaughlin Effect

3.1. HAT-P-49b: A Polar Hot Jupiter

3.1.1. Validating the Residual Velocity of HAT-P-49b

The initial data for HAT-P-49b implied a system with a large planet orbiting a hot, rapidly rotating star. Once the data on the transit night were collected, it was imperative to validate if the apparent blip in the radial velocity curve was a potential RM effect or if it was noise. With the orbital parameters for the system calculated, the radial velocity solution takes the form:

Vr .t/ V KRV Œcos .! f .t// e cos ! VRM .t/ D C C C C

V Initial Offset Velocity D (3.1) KRV Semi-amplitude Radial Velocity D

VRM .t/ Rossiter-McLaughlin effect D In Equation 3.1, ! is the argument of pericenter, f .t/ is the true anomaly, and e is the eccentricity of the system. The orbital parameters I used for HAT-P-49b are shown in Table 1. With the radial velocity fit, I could examine the residual velocity to look for theRM signal. The measurements are plotted with the orbital fit in Figure 3.1. Due to the large amount of scatter, I first needed to consider the significance of the residual velocity. The transit night observations produced three points before and three points after the transit with relatively little scatter. I compared the average of the residuals during transit with the average of the three points before ingress and three points after egress to ensure that the purported signal was something more than noise. Table 2 summarizes the averages and their respective standard deviation. Using a simple statistical test, the difference between the transit average and the averages outside of the curve was 3σ. While the curve was  not extremely significant, the signature seen in the radial velocity may have originated from – 34 – a perturbation to the radial velocity of the star. Comparing the signal on the transit night (blue) in Figure 3.1 to the examples in Figure 1.5, there appears to be very little symmetry in the signature for HAT-P-49b and this suggests it may be a polar planet.

3.1.2. Modeling the RM Effect in HAT-P-49b

The basic model used is seen in Equation 1.1 where I assumed the geometry described in Ohta et al. and assumed a quadratic limb-darkening law. I included a correction velocity to the model to account for any offset caused by an active stellar surface. It was suspected that HAT-P-49b had an active surface due to the large amount of scatter present in the radial velocity measurements. There was more scatter about the best-fit model than expected based on formal uncertainties, as such the jitter, a constant term, was added in quadrature to the radial velocity errors. The measurements from the FLWO had a mean jitter of 59:9 19:4 ˙ 1 1 [m s ] and those from the OHP had a mean jitter of 102:4 42:7 [m s ]. It is not known if ˙ the jitter was purely instrumental, astrophysical, or a combination of both. The addition of this constant term is similar to what Winn et al. (2006) did to their model for HD 189733. Winn et al. knew HD 189733 was chromospherically active and required a correction term to improve the fit. The astrophysical noise has times scales of days, which is much longerthan the duration of the transit. In the presence of this noise, the data points obtained on a single night will all be affected by errors which are strongly correlated. Stellar activity can range from solar spots to more violent events which may have lifetimes which are comparable to the orbital period of the star. It is possible to see large scatter among measurements if the surface flux of the star is constantly being perturbed. While it is not known what causesthe large jitter, it significant enough in HAT-P-49b to add a correction. I tried the model with a constant offset velocity,  :

Vr V KRV Œcos .! f / e cos ! VRM  (3.2) D C C C C C – 35 –

Fig. 3.1.— HAT-P-49b Radial Velocity Curve The radial velocity measurements are over-plotted along with the orbital solution in the top half of the graph. The functional form of the RV curve is in Equation 3.1 with VRM set to zero. The bottom half is the difference the residuals between the orbital fit andthe measurements. The data collected before October 18, 2013 are shown in red while the data after are shown in purple. The measurements on the transit night are in blue and appear to have less scatter because these points were collected on the same night. The data from the transit night were not used to derive orbital parameters. – 36 –

Table 1. Parameters for HAT-P-49b

Parameter Value

Light Curve Parametersa

Period [Days] 2:691548 0:000006 ˙ Tc [BJD] 2456399:624 0:0006 ˙ Tdur [hours] 4:11 0:05 ˙ a=R 5:1 0:3  ˙ Rp=R 0:079 0:002  ˙ b=R 0:34 0:14  ˙ i [degrees] 86:2 1:7 ˙ Limb-Darkening Coefficientsa

u1 (linear, fixed) 0:1003

u2 (quadratic, fixed) 0:3711 Radial Velocity Parametersb

1 KRV [m s ] 164:01 9:06 ˙ e 0 (fixed) 1 V [m s ] 174:5 4:6 ˙ 1 V sin i [km s ] 16 0:5  ˙

aParameters from Bieryla et al. (2014).

bParameters derived using the procedure repack_multi_cc_generic. – 37 –

I initially attempted to fit the radial velocity distortion with only the stellar rotational velocity, the offset velocity, and the obliquity as my free parameters. The resulting fitwasnot ideal and produced poor residuals (Figure 3.2). The model needed to be optimized for a more complex system which, in this case, meant that the out of transit slope of the fit needed to change. Looking at Equation 3.2, the amplitude of the radial velocity solution is determined by KRV . I decided to therefore set the semi-amplitude velocity, KRV , as a free parameter called KRM , in addition to  and Vı . I let V sin i be the free parameter called VRM; sin i .    

While it may seem a bit unorthodox to let KRV be a free parameter, Albrecht et al. (2012) provide some examples of the RM effect with such a model and provided an explanation

for why KRV can be allowed to change. If the stellar surface suffers from spurious noise of an astrophysical nature, such as an active stellar surface or sun spots, this may produce gradients in the radial velocity on timescales of a few hours. With prolonged noise we expect a perturbation in the radial velocity signal. The out-of-transit measurements for the system were already highly scattered, suggesting some type of astrophysical noise. While the noise of the star varies with age, in general dark spots or other variations can cause shifts in measurements of order 10 100 m/s (Lovis & Fischer 2011), a value which can significantly  alter the measured KRV .

I used the orbital parameters to solve for the normalized spatial coordinates X and

Table 2. The Average Offsets forT-P-49b HA

1 1 Time In Transit Average Vr [m s ] σ [m s ]

Three Points Before Ingress 19:73 5:42 In Transit Points 90:26 18:07 Three Points After Egress 53:12 8:38 – 38 –

Fig. 3.2.— HAT-P-49b Fit using orbital parameters The fit used above allowed the stellar rotational velocity, the obliquity, and the velocity offset to alter as free parameters. There is a noticeable in the residuals, indicated inblue the bottom panel. The same color scheme was used as in Figure 3.1. For the transit night only (22 light blue points), χ2 18:188 and a χ2 1:07 for 18 degrees of freedom. The D red D obliquity derived from this fit was  99ı.  – 39 –

Y for HAT-P-49b. The initial test of the system ignored limb-darkening and resulted in a poor fit. The linear limb-darkening model from Ohta et al. did very little to improve the model. The addition of a quadratic limb-darkening law produced a great improvement in the best-fit model. The fractional loss of light was calculated using the occultsmall procedure by Mandel & Agol to produce a synthetic light curve of HAT-P-49b. From the synthetic light curve, the fractional loss of light, ı, could be derived as a function of known parameters. To constrain the value of λ, the model was fit to the data by minimizing χ2. The spectrographic analysis produced by Bieryla et al. provided both radial velocities and errors in those measurements. IDL contains a procedure for the minimization of multidimensional functions known as amoeba (Nelder & Mead 1965; Press 2007). It is a robust downhill simplex minimization routine that allows for exploration of parameter space. The procedure is given a starting point from which small steps are taken in each variable to find the minimum of the function. For the amoeba procedure, χ2 was defined as:

N 2 .RVi; RVi; / 2 obs calc (3.3) D 2 i 1 .i / XD where N is the total number of radial velocity points, RVi;obs is the observed radial velocity, and RVi;calc is the calculated radial velocity from Equation 3.2, and i is the error of each respective measurement. For this system, I fixed most of the systemic parameters derived in the discovery paper. I let the RM velocity correction,  , the obliquity, λ, the semi-

amplitude velocity, KRM , and the projected stellar rotational velocity, VRM; sin i be free   parameters. The stellar rotational velocity was set as a free parameter to alter the amplitude of the RM effect. It is important to emphasize the KRM and VRM; sin i derived from   the model are specific only to the transit night and are not meant to be taken as general parameters for the system. I only allowed  to be a constant value and to only shift the points on the transit night.

I used χ2-fitting to provide the initial parameters for the model. I utilized the – 40 –

Markov Chain Monte Carlo algorithm to both generate the best-fit value and the posterior distribution for each parameter. The free parameters for the MCMC procedure were , V ,

  VRM; sin i , and KRM . For , I set a constraint of 2  2 to break the degeneracy     of the angle. My approximation for the RM effect is periodic with  and as seen in Figure

1.3 the value of  can move either clockwise or counterclockwise by 90ı. I ran a nine million chain MCMC to generate a smooth posterior distribution.

3.1.3. Results and Discussion for HAT-P-49b

From the MCMC algorithm, it was determined that HAT-P-49b had an obliquity of

 90:1 16:7ı. All of the other parameters derived from MCMC can be seen in Table  ˙ 3 with their respective confidence intervals. I obtained posterior distributions for the four

parameters: ,  , VRM; sin i , and KRM . The median value was accepted as the most   probable value and is given in Table 3. I used the posterior distribution to determine any potential correlation between  and the other free parameters. A two dimensional histogram of  and V sin i is seen in Figure 3.3. A high amount of correlation would suggest a   linear relationship between the two, which is not the case in actuality. While there is some

correlation between  and VRM; sin i , this is may arise from the nature of the observations,   or perhaps the geometry of the transit. All other correlation plots are in Section C.1 of the appendix.

The final fit for the transit night data is shown in Figure 3.4. The RM effect alone does not distinguish the direction of , clockwise or counterclockwise with respect to the stellar spin vector, in which the planetary orbit is oriented. It does, however, say something about how HAT-P-49b could have been formed. Even with a rather large dispersion, the fit gives an obliquity that is far from zero, such that HAT-P-49b cannot be a coplanar object. This suggests that HAT-P-49b is a polar body orbiting its 1:5 M F-type host. If HAT-P-49b ˇ – 41 – is compared to the other planets in Figure 1.4, we see it seems to agree very well with the trend that hot Jupiters around hotter stars have higher obliquities. Given the value of , the formation of HAT-P-49b could not have been due to a process which retains coplanarity. Instead, assuming the star is aligned with the protoplanetary disc, it may have been swung inward from one of the more dynamically active mechanisms.

3.2. KELT-7b: An Aligned Hot Jupiter

3.2.1. Determining the Transit Parameters for KELT-7b

The KELT-7 system, unlike HAT-P-49 , does not have a discovery paper and most of its parameters have not yet defined. I could fit some of the orbital parameters usingSPC but proper analysis required the parameters from the light curve. I had four separate light curves for the system. One was from Nov. 2012 while the rest were in 2013 on Sept. 19, Sept. 20, and on the transit night of October 19. I used TAP to produce the necessary system parameters for the RM Effect analysis. TAP fits the given light curve data using MCMC. I initialized the analysis of KELT-7b with the estimates of the orbital parameters from repack_multi_cc_generic. I did not allow the period or Tc [BJD] to vary because

Table 3. MCMC Fitting Parameters for HAT-P-49b

Parameter Physical Significance Value Correlation Coefficient

 [degree] sky-projected obliquity 90:5 14:5 ——– ˙ 1 15:6  [m s ] velocity offset, RM specific 227:3C17:5 0:012 1 61:3 KRM [m s ] out-of-transit slope, RM specific 64:1C63:7 0:019 1 12:1 VRM; sin i [km s ] amplitude, RM specific 20:9C10:6 0:012   – 42 –

Fig. 3.3.— HAT-P-49b Correlation between λ and VRM; sin i   The 1 confidence intervals for the posterior distribution generated via MCMC are indicated by the dashed lines. Within the demarcated interval of each posterior distribution, lay roughly 68% of the chains from MCMC. To check for potential correlation among my free parameters, I created two dimensional histograms. The different σ levels, indicated by the red contours, correspond to where 68%, 95%, and 99:7% of the data can be found. – 43 –

Fig. 3.4.— HAT-P-49b Fit From MCMC The fit used above allowed the stellar rotational velocity, VRM; sin i , the obliquity, , the   velocity offset,  , and the semi-major amplitude, KRM , to alter as free parameters. The values of the parameters used for the fit are seen in Table 3. Compared to Figure 3.2, the residuals have a smaller absolute value. For the fit above, χ2 177:06 and χ2 10:42 for D red D 18 degrees of freedom. The fit corresponds to an obliquity of  90ı.  – 44 –

I assumed they were known precisely enough through photometry that their uncertainties would have no effect on the other parameters. I assumed a circular orbit, fixed e to 0 and used TAP to fit for limb-darkening parameters. The limb-darkening parameters were allowed to evolve in a such a way that the limb-darkening was positive and decreasing towards the center of the star (Gazak et al. 2012). I ran a total of five separate chains with a length of five million states. The five chains were combined to find the values listed inTable 4. To test the veracity of the transit fit, I made a synthetic light curve for KELT-7b and asseen in Figure 3.5, it coincided fairly well with the transit night light curve. This synthetic light curve was used to generate the fractional loss of light necessary for the RM model.

Data Fitted Light Curve Residuals

1.00

0.99

0.98 Relative Flux (i band)

0.97

0.96

0.04 0.02 0.00 0.02 0.04 Phase

Fig. 3.5.— Synthetic Light Curve for KELT-7b The lightcurve for the transit night on October 18, 2013 was made using the IDL program occultsmall by Mandel & Agol. The duration of the event is roughly 3:5 hours long. The residuals are plotted in red. The depth of the transit was roughly 0:09. I used the three light curves for KELT-7b that were available when fitting with repack_multi_cc_generic. The resulting orbital parameters were then used to make the synthetic light curve. – 45 –

Fig. 3.6.— KELT-7b Radial Velocity Curve The radial velocity measurements are over-plotted along with the orbital solution in the top half of the graph. The bottom half shows the residuals between the fit and the measurements. The data collected before October 19, 2013 is shown in red while the data on the transit night are green. Note the difference between this sharp symmetric signature and that ofthe HAT-P-49b in Figure 3.1. While the radial velocity solution may not appear to be a good fit, David Latham’s group has collected data up to February 2014. The radial velocity solution is much more believable with the newer data. – 46 –

Table 4. Parameters for KELT-7b

Parameter Value

Light Curve Parameters

Period [Days] 2:734775

Tc [BJD] 2455958:684

Tdur [hours] 3:559 0:98 a=R 4:6C0:79  Rp=R 0:09 0:008  ˙ b=R 0:5 0:3  ˙ 4:1 i [degrees] 82:4C3:2 Limb-Darkening Coefficients

u1 (linear) 0:6 0:4 ˙ u2 (quadratic) 0:05 0:4 ˙ Radial Velocity Parametersa

1 KRV [m s ] 135 26 ˙ e 0 (fixed) 1 V [m s ] 92:9 16:8 ˙ 1 V sin i [km s ] 73:1 0:6  ˙

aParameters derived using repack_multi_cc_generic. – 47 –

3.2.2. Modeling the RM Effect in KELT-7b

The data on the transit night for KELT-7b had a more noticeable signature than HAT- P-49b (Figure 3.5) due to the much larger stellar rotational velocity. Looking at the radial velocity measurements, there was a distinct and classic RM blip for a nearly aligned planet. The base radial velocity solution was given by Equation 3.2. While the data outside of the transit had a lot of scatter, the data on the transit night did not. As in the case of HAT- P-49b, the large amount of scatter in the radial velocity measurements suggested stellar activity and adoption of a model where I had four free parameters.

Aside from the slight change in model, the rest of the procedure was exactly the same as for HAT-P-49b. I assumed a quadratic limb-darkening law and constrained the value of λ through minimization of χ2 using IDL’s amoeba and with χ2 defined as in Equation 3.3. To determine the errors in the fitted parameters I utilized the MCMC algorithm. The Markov chain was terminated after nine million steps.

3.2.3. Results and Discussion for KELT-7b

2 From the χ -fitting of KELT-7b, an obliquity of λ 8 17ı was derived. All other  ˙ relevant parameters required to fit this system can be seen in Table 5. Figure 3.7 shows the correlation between  and VRM; sin i along with their respective posterior distributions.   All other correlation plots and posterior distributions can be seen in Section C.2 of the appendix. For all of the distributions, the median value was taken as the best fit. The final fit using the MCMC parameters can be seen in Figure 3.8. The small angle combined with the relatively small confidence intervals (smaller than 90ı) ensure that KELT-7b cannot possibly be misaligned in a similar fashion as HAT-P-49b. Figure 3.9 offers a glimpse at the projected geometry the calculations in this paper suggest. This does not necessarily say – 48 – anything about the formation of KELT-7b. It may very well have formed through many body interactions only to circularize with enough time. It is a bit peculiar to see a well aligned planet with a hot host star and KELT-7b would be an anomaly on the temperature-obliquity graph in Figure 1.4. The obliquity derived for KELT-7b is simply a first-order approximation. I would to see more data for KELT-7b to better constrain the orbital solution and I would also like to see another RM effect. David Latham’s group is currently performing follow-up observations on KELT-7b for a discovery paper in progress.

3.3. Improved Modeling of the RM Effect

I decided to use Hirano’s latest analytic method (Hirano et al. 2011) to derive the obliquity and compare it to the above results. A detailed derivation of the analytic model is presented by Hirano et al. in that paper. The improved analytic method includes more realistic stellar absorption line profiles, differential rotation, macro/micro-turbulence, thermal broadening, pressure broadening, and instrumental broadening. To call upon Hirano’s model, I used a routine written in C, called new_analytic7, provided by Roberto Sanchis-Ojeda at MIT. The routine requires linear and quadratic limb darkening coefficients

,u1 and u2, the equatorial rotational velocity, V sin i , the macroturbulence dispersion, , the  

Table 5. MCMC Fitting Parameters for KELT-7b

Parameter Value Correlation Coefficient

 [degree] 8:3 16:8 ——– ˙  [m s 1] 71 51 0:717 ˙ 1 KRM [m s ] 466:2 247:2 0:33 ˙ 1 VRM; sin i [km s ] 51:1 13:1 0:114   ˙ – 49 –

Fig. 3.7.— KELT-7b Correlation between λ and VRM; sin i   As with HAT-P-49b, I decided to check for correlation among my parameters. The different σ levels, indicated by the red contours, correspond to where 68%, 95%, and 99:7% of the data can be found. Compared to Figure 3.3, the obliquity seems to be fairly uncorrelated in this case. – 50 –

Fig. 3.8.— KELT-7b Fit From MCMC The fit used above allowed the amplitude, VRM; sin i , the obliquity, , the velocity offset,    , and the out-of-transit slope, KRM , to alter as free parameters. The values of the parameters used for the fit are seen in Table 5. For the fit above, χ2 30:578 or χ2 =1.33 D red for 24 degrees of freedom. The fit corresponds to an obliquity of  8:1ı. One should note  that the shape of an aligned system such as KELT-7b is much different from that of a polar planet (Figure 3.4). The points at the end of the transit (phase > 0:02), fit poorly to the solution because they were contaminated by dawn. – 51 –

Fig. 3.9.— The systemic geometry of HAT-P-49b and KELT-7b A. The apparent geometry for HAT-P-49b is shown using the calculated obliquity  90ı.  This is how the system would look from the vantage point of us, the observer. It is not possible to extract the three dimensional geometry from the obliquity alone. The obliquity is indicated as the black angle. The spin vector of the planetary orbit is the cyan vector and the stellar spin orbit is the orange vector. B. The geometry for KELT-7b using the calculated obliquity of  8ı. The rotation of  KELT-7 is indicated by the green circle. Each system was parameterized in Mathematica. – 52 –

Rp coefficient of differential rotation, ˛, cos i, the ratio R , the Gaussian dispersion of spectral  lines, ˇ, the Lorentzian dispersion of spectral lines, , and the convective blueshift (CB). ˛ describes the degree of differential rotation present in the star. ˇ is a fixed parameter that is assumed to be proportional to the width of the instrumental profile. Its true value is given as: 2k T ˇ B eff 2 ˇ2 D s  C C IP 2 where  is the reduced mass of the molecule,  is the microturbulence dispersion, and ˇIP is the dispersion of the Gaussian which best fits the instrumental profile. Hirano et al. (2011) states there is a strong degeneracy between and ˇ, as such once ˇ is fixed the value of can be approximated from the line profile. Each spectral line has a unique value for ˇ and for and it is simpler to give the effective values via comparison of the analytic formula with simulated results. Hirano et al. (2011) give a range of potential values for each of the spectral profile parameters. I used the parameters recommended for solar-like stars:  4 D 1 1 1 km s , ˛ 0:2, ˇ 3 km s , 1 km s , and I used a solar convective blueshift of D  D 0:5. The results from MCMC are shown in Figures 3.10 and 3.11. The resulting fits for the transit night are seen in Figures 3.12 and 3.13. From fitting with Hirano’s analytic method,

HAT-P-49b had an estimated  99 10ı and KELT-7b had an estimated  1 10ı.  ˙  ˙ Both of these calculations are in agreement with the results in previous sections of this thesis. HAT-P-49b is still a polar planet and KELT-7b is still aligned. – 53 –

1.0 = 99.1 ± 10.4 [deg]

0.8

0.6

0.4

0.2 Probability Distribution Function 60 ± V*sini= 20.0 6.9 [km/s] 50

40

30 sini [km/s] *

V 20

10

0 60 80 100 120 140 160 180 0.2 0.4 0.6 0.8 1.0 λ [degrees] Probability Distribution Function

Fig. 3.10.— MCMC result for HAT-P-49b using Hirano’s analytic method I used the analytic method developed by Hirano et al. (2011) to calculate the obliquity. The figure above shows the correlation between  and VRM; sin i . The estimated obliquity from   MCMC is  99 10ı. While this is not my exact result, given the errors in this calculation  ˙ and the previous errors, the values are comparable. They are in agreement with HAT-P-49b being a polar planet.

Fig. 3.11.— MCMC result for KELT-7b using Hirano’s analytic method The figure above shows the correlation between  and VRM; sin i . The estimated obliquity   from MCMC is  1 10ı. The estimation is in agreement with my previous result, and  ˙ suggests KELT-7b is aligned with its host star. – 54 –

Fig. 3.12.— HAT-P-49b Fit using Hirano’s analytic method fit above is made using the parameters derived from the MCMC and Hirano’s analytic expression. The velocity offset,  , and the out-of-transit slope, KRM , were derived using χ2 fitting. The residuals for the transit night event are shown in the bottom panel. The χ2 43:25 and the χ2 2:54 for 17 degrees of freedom. D red D

Fig. 3.13.— KELT-7b Fit using Hirano’s analytic method fit above is made using the parameters derived from the MCMC and Hirano’s analytic expression. The velocity offset,  , and the out-of-transit slope, KRM , were derived using χ2 fitting. The residuals for the transit night event are shown in the bottom panel. The χ2 56:37 and the χ2 2:09 for 24 degrees of freedom. D red D – 55 –

4. Constraining the Obliquity with Stellar & Transit Parameters

4.1. Required Parameters

Analysis of the RM effect can be powerful but, it is ultimately limited to bright host stars with large transiting planets. The next part of this thesis will attempt to constrain the stellar inclination i in an attempt to potentially misaligned systems with an inclination  difference. This approach instead relies on the stellar parameters to derive stellar inclination. The true angle between the spin vector of the orbital plane and the rotational axis of the star star (Figure 1.9) is a function of the inclinaton of the stellar rotational axis, i , the  inclination of the orbital plane inclination, ip, and the sky-projected obliquity, . If I focus only on planets confirmed through the transit method, then we know ip 90ı. As discussed  in Section 1.3.2, since ip is constrained by nature of the transiting population, only i  has to be constrained to find limits for the obliquity. The promise with this methodis that it can identify potentially misaligned systems which can be later analyzed and models more rigorously. If there is a large database of planet harboring stars, such as Kepler, it is possible to quickly sift through the data and identify planets with disrupted geometries. For my analysis, I utilize the stellar parameters from the Kepler mission, with line broadening measurements from TRES spectra, and periods from McQuillan et al. (2013).

4.1.1. Stellar Parameters

The NASA Exoplanet Archive has compiled an enormous list of 3,845 Kepler candidates as of this writing. All the Kepler Objects of Interest (KOIs) are listed with fundamental

stellar properties: mass, M , temperature, T , radius, R , effective surface temperature, Teff,    surface gravity, log g, and their density, . For the purposes of constraining , I only needed the stellar radii. I made use of the most recent data set available from Huber et al. (2014). – 56 –

The five sources of data considered in the article were either asteroseismology, transits, spectroscopy, photometry, and the Kepler Input Catalog, KIC. For the most recent catalog on the Exoplanet Archive, Huber et al. adopted the 2012 isochrones from the Dartmouth Stellar Evolution Database (DSEP). After interpolation, only the models with solar-scaled ˛-element abundances were used for fitting. Fourth order polynomials were fit between isochrones which were compared to atmospheric models (BT-Settl) to derive typical stellar properties. The radii in the database had a offset of 1% and an scatter of 7% when compared to published data for stars. Temperatures were derived by comparing observed colors to theoretical models from DSEP and adopting the same reddening model in KIC. I adopted these values for the radius and temperature in my analysis.

4.1.2. Spectroscopic Rotational Velocities

The stellar rotational velocity measurements used data from Keck HIRES (High Resolution Echelle Spectrometer), FIES (Fiber-fed Echelle Spectrograph) at the Nordic Optical Telescope, TRES. and McDonald spectra. The SPC procedure was used to analyze the spectra and produce the rotational velocity. For multiple observations, the calculated V sin i is the weighted average using the normalized cross-correlation function (CCF) peak   height. To limit the use of poor observations a CCF peak height limit was placed. The CCF limit for these values was 0:9. Most of the CCF were larger than 0:9 but a few were also  under. Despite the use of the weighted average for V sin i , poor classifications often have   a CCF height of 0:8 and can strongly affect the final SPC result. For this paper, Ihad  a total of 407 rotational velocities which were calculated by Buchhave et al. (in press in Nature). – 57 –

4.1.3. Stellar Rotational Period

One set of rotational periods originate from the work of McQuillan et al. (2013). The measurements were made using an autocorrelation function (ACF) method. It measures the degree of self-similarity among the lightcurves over a range of lags. Repeated star spot crossings cause the ACF to peak at lags which are integer multiples of the rotational period. McQuillan et al. analyzed 1919 exoplanet main-sequence host stars and determined rotational periods for 737 KOIs. They ignored confirmed and potential eclipsing binaries. Data from quarters 0 and 1 were omitted due to the short duration of observation and data from quarter 2 were not used due to residual systematics. The data from quarters 3 through 14 were corrected using PDC-MAP (Presearch Data Conditioning-Maximum A Posteriori) (Smith et al. 2012), a pipeline that removes any instrumental trends by fitting cotrending basis vectors to represent common non-astrophysical trends. The periodic variability for the stars were compared to photometric observations and they found general consistency between the published rotational period and the period derived from previous observations. Of the 407 KOIs for which I had rotational velocities, only 125 had rotational periods from McQuillan et al.

The other set of rotational periods is from the work of Walkowicz & Basri (2013). The authors only used long cadence data which were reduced using PDC-MAP. Lomb-Scargle periodograms (Scargle 1982) and variability statistics (Basri et al. 2010) were computed for all stars. Walkowicz & Basri mentioned that if the data are of high quality and nearly continuous, various significant peaks are present. Some of these peaks in the periodogram represent the true rotational period while other peaks are harmonics. While they took extensive measures to ensure only those with a power of 800 and a period of less than 45 days were analyzed, if a star has symmetric magnetic features on opposite hemispheres, the period would appear to be half of its true value. Stars which met the criteria had their light – 58 – curve examined by-eye minimize confirm the periods found. Of the 407 KOIs for which I had rotational velocities, only 83 had rotatioal periods from Walkowicz & Basri.

4.2. Results

The final results for the stellar inclination revealed the naivety of this approach. The notion of constraining stellar inclination from stellar parameters is ultimately dependent on the accuracy with which we measure the rotational period, the stellar radii, and the rotational velocity. When I used Equation 1.2, the majority of sin i were impossible. For  the Walkowicz rotational periods, 41 of the 83 KOIs had sin i 1 while for the McQuillan   rotational periods, only 38 of the 125 KOIs had sin i which were possible. For some KOIs  the reported sin i was as high as 5, something which indicates one or all of the parameters   are inaccurate. These results for the McQuillan rotational periods are seen in Figure 4.1 and those for the Walkowicz rotational periods are seen in Figure 4.2. Even if I included the reported errors in measurements, for most of the stars there was an insignificant change in the inclination. I was unable to make any general conclusions using stellar parameters because so much of the data seemed to give impossible results. Complications arise when the observed line broadening is larger than can be predicted by the radius and period of rotation of a star. The rotational broadening will increase if the period is shortened or if the radius is larger than the actual value. In the following sections I will attempt to address the potential for errors in either radius, rotational period, or rotational velocity.

4.2.1. Potential Error in the Stellar Radius

The value of sin i is inversely proportional to the stellar radius. In order to double the  value of sin i the stellar radius would need to be halved. The radius from Huber et al. are  – 59 –

Fig. 4.1.— Stellar Inclination from McQuillan Rotational Periods The figure above shows the value of sin i and i for the KOIs that had a rotational period   from McQuillan et al. (2013). The top panel shows the sin i for all 125 KOIs. The dashed  line delineates the physical limits of the value. Red stars are those where Teff > 6250 K, a cut-off temperature which Albrecht et al. (2012) noted as having a high occurrence of oblique exoplanets. Following the assumption that transiting exoplanets are often seen head-on, a system with an oblique planet should have a low value of sin i . The bottom panel shows the  calculated i for the stars which had physically possible sin i . Given the impossible values   for sin i , general trends in obliquity cannot be determined.  – 60 –

Fig. 4.2.— Stellar Inclination from Walkowicz Rotational Periods The figure is in the same layout as Figure 4.1 but with rotational periods from Walkowicz & Basri (2013). The top panel shows all of the 83 KOIs and their sin i while the bottom  panel shows the physically plausible stellar inclinations. Stars with Teff >6500 K are shown in red. – 61 – noted to be heavily dependent on the surface gravity computed for the host star. There is an inherent difficulty in estimating stellar parameters, particularly the surface gravity and metallicity, from broadband colors. To overcome any potential degeneracies in matching broadband colors to models while estimating parameters, Huber et al. adopted a metallicity prior based on the distribution of the Geneva-Copenhagen survey, a T log g prior based eff on the number density in Hipparcos, and a prior based on the number density of stars as a function of galactic latitude. Despite all of the efforts to reduce potential error, Kepler Input Catalogue (KIC) surface gravities are assumed to be accurate only to 0:5 dex and  are often overestimed (Verner et al. 2011; Everett et al. 2013). The lower surface gravities were measured via asteroseismic methods on Kepler light curve data for 514 solar-type stars by Verner et al. and by the spectroscopic study on 352 stars from the KIC database by Everett et al.. The measurements were comparable to each other and, on average, for stars with log g > 4:0 dex the measured surface gravities were 0:23 dex lower than the results in KIC. This can underestimate the radii by up to 50%. Verner et al. did mention a potential for bias in the asteroseismic log g due to Malmquist bias, which would result in the selection of more evolved, bright stars with higher amplitude oscillations often associated with a lower surface gravity.

This potential for bias in observation was explored by Gaidos & Mann (2013). An observational bias may imply the fraction of sub-giant stars in the Kepler samples are underestimated. There is an apparent magnitude limit to the Kepler mission which may result from the signal-to-noise ratio requirement or the necessity to confirm transiting systems via Doppler observations. Such a magnitude limit would favor stars with greater luminosity. These brighter stars can be included at larger distances, increasing the volume of space of the survey and a predisposition for higher luminosity. The Malmquist bias results from this increase in sampling volume. The probability of a transit detection is directly proportional to the stellar radius which, at a given temperature, scales with the distance probed. The – 62 – luminosity and radius relationship suggests that a bias towards brighter stars results in a bias toward larger stars. The uncertainty in stellar radius for KOI host stars can be seen in Figure 4.3. The large uncertainty in the KIC measurements may be attributed to errors in photometry, degeneracies between stellar parameters and color, and even errors in the models themselves. Brown et al. (2011) conclude that for stars where Teff > 5400 K, no information on surface gravity or radius could be accurately discerned from the data. Stars where T 5400 K are in agreement with parameters derived from asteroseismic eff  measurements and seem the distinction between main-sequence stars and giants appears to be reliable with a confidence of 98%(Brown et al. 2011).

Fig. 4.3.— The Uncertainty in Stellar Radius The figure shows the uncertainty in the radius of KOI host stars. The uncertainty isdefined as half of the range containing 68% of the probability distribution of the radius. The open points are giant stars while the dots are dwarfs. High temperature stars have a larger uncertainty. Malmquist bias is thought to play a significant role in this error. This image is from Gaidos & Mann (2013). – 63 –

4.2.2. Potential Error in the Rotational Period

I already mentioned the limitations with the use of Lomb-Scargle periodograms in the analysis by Walkowicz & Basri. While they did take extensive measures to prevent recording harmonic periods, there is always the possibility some of the periods may be half of its original value. However, a more significant error in calculation affecting both McQuillan and Walkowicz may be the use of the PDC-MAP pipeline on long-period stars. When using the PDC-MAP pipeline, Kepler light curves are decomposed into three separate length- scales to which the standard pipeline is independently applied. A subset of highly correlated and “quiet” stars, those without magnetic activity and star spots, are used to obtain the cotrending basis vectors (García et al. 2013). These vectors establish the range of reasonably robust fit parameters for instrumental features present in the data. In the band withthe longest timescale, with periodicity greater than 21 days, the light curves are fit to the cotrending basis vectors. For all PDC-MAP analysis, any signal longer than 21 days is effectively removed (García et al. 2013) because it is assumed that the stellar signal can no longer be distinguished from any systematic instrumental effects. With any long period signatures, there is the possibility of masking the true value and instead reporting a smaller harmonic period.

García et al. is an example of the potential error when using the PDC-MAP pipeline with KIC 11026764, known as Gemma (Figure 4.4). The average surface period, determined with improved algorithms, is 33 days (Mathur & Lynas-Gray 2013). A high-pass filter of the pipeline completely removes the signature at of its period. The result is a power spectrum with the highest peak at 12:5 days, suggesting this is the actual period. The smaller value is simply a harmonic of the original period. The lower frequency signals in PDC-MAP may not be the true period and harmonics of the actual value are always something to be wary of when trying to deduce rotational periods. – 64 –

Fig. 4.4.— Harmonics and the PDC-MAP Pipeline for Gemma I. The power spectrum was computed using methods from Smith et al. (2012) and Stumpe et al. (2012). The use of a recent variant of the pipeline, PDC-“multi-scale”-MAP, erases any signature longer than 21 days. The result is a power spectrum with a peak at one of the harmonics of the period at 12:5 days. This gives a severe underestimate of the true rotational period. II. Analysis of the same data with a different correction developed by the Kepler Asteroseismology Science Consortium, produced a power spectrum with the true value of the rotational period as 33 days. The method allows for correction of outliers and combination of data from different quarters without the use of a high-pass filter which would otherwise remove the signal. Both images are from a presentation given by R.A. Garcia at the 2013 International Francqui Symposium titled “Asteroseismic Measurements: A New Way To Study Rotation And Magnetism In Stars”. – 65 –

4.2.3. Potential Error in the Rotational Velocity

The rotational velocities were calculated using SPC 2.7, a program which depends on cross-correlation methods. An observed spectra is matched to a library grid of synthetic model spectra and is extended to yield values whic hare not constrainted to grid points Buchhave et al. (2012). The temperature, surface gravity, metallicity, and rotational velocity are altered as the spectrum is cross-correlated to the library of synthetic spectra. The

1 rotational velocity for the grid is varies from 0 < V sin i < 200 in km s with a spacing   1 that varies from 1 20 km s . The library contains a total of 51359 synthetic spectra. Each synthetic spectra is cross correlated to the observed spectra and the peak values of the normalized cross correlation function are fitted with a three dimensional surface. The peak of the surface is the value for the parameter. This cross correlation method assumed that the rotational velocity is weakly correlated with the other three parameters.

The cross-correlation method is not useful if there are no well-defined features in the spectrum. The error in rotational velocity also becomes significant for slow rotators, typically stars with velocities less than the instrumental resolution. Slowly rotating stars have line widths close to the limits imposed by spectral resolution (Díaz et al. 2011). The spectral

1 resolution for TrES is typically approximately 6:8 km s . The measurement of these lines are more difficult for early-type stars because of the weaker metallic lines(Díaz et al. 2011). A plot of the velocities for each respective dataset can be seen in Figures 4.5 and 4.6. Most of the KOIs examined were relatively slow rotators and there is always the possibility that an overestimate in the true value may have contributed to an an overestimate of the line

1 broadening. Both plots show a band of velocities from 2 4 km s which span sin i values   of 1 3. From a first gland, however, it is difficult to see a general trend. The observed line broadening has more components than the expected line broadening, where I assumed most of it was due to rotational broadening. Some example of other sources – 66 – of broadening which are not included in SPC are macroturbulence and microturbulence. Macroturbulence is assumed to operate on large scales in the star and is considered to correspond to surface granulation or convective cells (Gray 2005). Microturbulence corresponds to small-scale motion where the characteristic length is small compared to the photon mean free path. The small scale motions produce Doppler shifts which are analogous to those arising from thermal broadening Gray (2005).

Microturbulence increases as surface gravity decreases and is therefore more significant for giant and subgiant stars (Edmunds 1978). Late-type main sequence stars instead

1 have microturbulent velocities in the range of 0:2 1:5 km s (Gray et al. 2001).  Macroturbulence is also shown to correlate with spectral type. For the sun, the macro

1 turbulent velocity is on the order of 3 4 km s while for F-type stars it is larger. The  macroturbulent velocity has been shown increase with increasing luminosity, and presumably, a decrease in the surface gravity (Gray 2005). While the microturbulent velocity for main sequence stars is not large, macroturbulence does seem more significant. To improve the constraint on the line broadening it may be beneficial to derive velocities by including turbulent broadening.

5. Conclusion

5.1. Obliquity and the RM Effect

In this thesis the effectiveness of the RM effect was shown in the planetary systems HAT-P-49b and KELT-7b. These exoplanets, despite having a similar F-type host, differ by a significant amount in geometry. I emphasize that the model employed in this work, while informative, is fairly naive. In addition, HAT-P-49b and KELT-7b need more observations to lower the scatter outside of transit and to improve the RM signature. While it is impossible to – 67 –

Fig. 4.5.— The relationship between sin i and velocity with the McQuillan dataset  Above is a plot of the calculated sin i using the rotational period from McQuillan et al.  (2013). There is wide range in the velocities of the KOIs. The impossible values of sin i  span the entire range of rotational velocities.

Fig. 4.6.— The relationship between sin i and velocity with the Walkowicz dataset  Above is a plot of the calculated sin i using the rotational period from Walkowicz & Basri  (2013). There are quite a few slow rotators, making it difficult to properly constrain the rotational velocity with small uncertainty. Unlike the McQuillan dataset, most of the impossible sin i have relatively larger velocities.  – 68 – make any generalization about a hot Jupiter formation from a sample of two systems, gaining knowledge of obliquity can immediately rule out certain formation ideas. In particular, for polar planets such as HAT-P-49b, migration may not be a result of planet-gas interactions. Migration may have been the work of a more violent and random interaction with another body. For hot Jupiters and other gaseous planets, the we effectively get a snapshot of the systemic geometry and, by extension, their dynamical history.

5.2. Obliquity and Stellar Parameters

The large number of cases where the observed spectroscopic line broadening was greater than the calculated equatorial rotational velocity, V sin i , prevent the examination of any   general trends in the obliquity. The expected equatorial velocity was calculated using orbital period and radius estimates for the host stars of Kepler transiting planet candidates. While it appears to be a simple calculation to determine the stellar inclination, it becomes difficult when the data itself are not perfect. Even if the error estimates for the data are included, it does not rescue the explain the large values observed for line broadening. It is also non-trivial to determine from which parameter the most significant error originates. The results obtained in this thesis may be the result of errors in one of the measurements or any combination. Hirano et al. mentioned they derived the stellar radius independent of photometric data, owing to the fact that it is impossible to constrain a proper ratio between semi-major axis and stellar radius without knowledge of the eccentricity. Instead of using the transit depth to approximate the stellar radius, they employed the Yonsei-Yale isochrone model to estimate radius. This method was not expected to be as accurate as the analysis of the RM effect. It was merely intended to reveal systems which may potentially be misaligned and then further studies through the RM effect to determine the value of . While Hirano et al. managed to get some results with this method, I think the potential for errors make – 69 – it rather difficult to achieve consistent and reliable results in exoplanetary geometry. Direct analysis with the RM effect or with asteroseismology, while not as simple as using stellar parameters, can provide richer information about any system.

5.3. The Obliquity and Exoplanetary Formation

The obliquity allows astronomers to identify some aspect of geometry for the system. I emphasize this is not the true geometry of the system, but merely the projected geometry apparent to the observer. To know the true geometry of an exoplanetary system would require knowledge of the three-dimensional obliquity, . However, the projected obliquity, , is can help determine if the exoplanet is aligned with its host star. Of the 79 spin- orbit alignments listed in the Holt-Rossiter-McLaughlin Encyclopedia, roughly 33 have a significant misalignment greater than 22:5ı. The substantial degree of misalignment suggests some dynamic process is responsible for bringing these hot Jupiters near their stars. HAT- P-49b is an example of a misaligned hot Jupiter and it could not have been formed from interaction with the gaseous planetary disc. On the contrary, KELT-7b is aligned with its host star and it’s formation may have been a result of interactions with the disc or it may by tidal interactions with its host star.

It is impossible to conclusively exclude any formation theories given the limited number of systems with a measured . To effectively eliminate a theory requires a larger sample size and ideally a variety of exoplanets. As discussed in the introduction, hot Jupiters are easily probed with the Rossiter-McLaughlin effect. It is harder, though not impossible, for to calculate  for Neptune-like planets. There is potential for other techniques, such as asteroseismology, to identify the alignment of Earth-like planets. Only with such data would it be feasible to select between theories which suggest the star is misaligned from the protoplanetary disc or those which suggest it is the individual planets which are misaligned. – 70 –

To use the method, I assume the stellar parameters are well constrained and this is not the case for all stars. The errors in the input parameters are perhaps its biggest drawback. Without careful consideration for the parameters, this method is not informative. Another setback is that there must be an estimate for the rotational period, line broadening, and radius. Most of the KOIs from TrES do not have a calculated rotational period. The stellar surface must be active enough to give such an estimate and this currently limits the systems I can analyze. However, with the bulk of Kepler data available, this method may prove useful to identify systems for closer analysis via the RM effect or asteroseismology once the stellar parameters are well defined.

In this thesis I have calculated the spin-orbit alignment for two exoplanetary systems. For each system, much work remains in examining subsequent transits to improve the confidence interval for the values derived in this work. Both systems have the potentialto provide some information regarding the formation of hot Jupiters. Of particular interest is KELT-7b, which is aligned despite being near a star with presumably little tidal dissipation. This work has provided an example of only a fraction of the techniques proposed to calculate the obliquity. I look forward to seeing what the Kepler data can reveal about the obliquity of exoplanets and how this can be used to explore planetary formation dynamics. – 71 –

Appendix

A. The Radial Velocity Solution

The basic expression for the radial velocity curve without the signature from the RM effect is shown in Equation A1 below.

Vr .t/ V K Œcos .! f .t// e cos ! (A1) D C C C

The above expression is ultimately a function of time. I will present a brief overview of orbital elements to help the reader understand the nature of the form adopted for the RV solution. For an in-depth analysis of orbital elements, I direct the reader to Solar System Dynamics by Murray & Dermott.

A.1. Orbital Elements

Orbital elements are parameters used when considering the two-body problem. The motion of a planet is most simply analyzed in the rest frame of the host star. It is often easier to transform into a three-dimensional Cartesian coordinate system where the x-axis is taken as the major axis of the ellipse in the direction of pericenter, the y-axis is perpendicular and lies in the orbital planet of the planet, and the z-axis is the normal to the orbital plane. Figure A.1 shows the orbital motion with respect to the reference plan in three-dimensional space. The six fundamental elements that describe the orbiting planet are: the semi-major axis a, the eccentricity, e, the orbital inclination, i, the longitude of ascending node, , the argument of pericenter, !, the mean anomaly, M . The mean anomaly defines the position of the planet on the orbit as a function of time. If the mass and period of the bodies are known in addition to these six elements, the motion of the body in space can be derived. – 72 –

Fig. A.1.— The Orbital Motion in Three Dimensions In the figure above, the motion of a body in three dimensions is shown, only thesegment of the orbit above the red reference plane is shaded. The ascending node is the point on the reference plane where the orbit crosses. The angle between the reference direction and the line of the ascending node is . The angle between the line of the ascending node and closest approach is !. i is the inclination of the orbit relative to the reference plane. The transformed Cartesian orbit relative to the orbit is indicated by the small vectors. – 73 –

A.2. Deriving the Radial Velocity Solution

For analysis,  is taken to be zero. The mean anomaly, M n .t /, is a 2-π periodic D function that is linearly dependent on time and angular velocity, n. The eccentric anomaly, E, is related to the mean anomaly with M E e sin E. The eccentric anomaly is the angle D between the major axis of the ellipse and the radius from the center to the intersection point on a circumscribed circle (Figure A.2). The eccentric anomaly in turn relates to the true anomaly, f . It can be shown (Murray & Dermott 1999) that the radius in orbital elements

a Circumscribed Circle

Ellipse F ' O F _ y a r E f _ O x F

Fig. A.2.— The Relationship Between E and f The left side of the figure shows the circumscribed circle with an elliptical orbit, thetwo focii are indicated. The right side of the figure shows the general relationship between the eccentric anomaly and the true anomaly for the orbit of the green object.

2 a.1 e / e sin f r r rf for a planetary system is 1 e cos f and the velocity is P 1 e cos f . By Kepler’s second D C P D C law: dA 1 r 2f ; integrating over one period dt D 2 P A 1 r 2f , substituting the area of an ellipse P D 2 P (A2) 2a2p1 e2 2a .1 e cos f / rf C P D rP D P p1 e2 – 74 –

If the value of r is substituted into the expression for r: P P e sin f r rf P D P1 e cos f C (A3) 2ae sin f D P p1 e2 The general expression for the radial velocity along the observer’s line of sight is vr D rf cos .f !/ r sin .f !/ sin i v0. If I substitute the expression for r and rf ,I P C CP C C P derive the expression used in this thesis: 2a .1 e cos f / 2ae sin f vr C cos .f !/ sin .f !/ sin i v0 D p 2 C C p 2 C C  P 1 e P 1 e  2a sin i Œcos .f !/ e cos f cos .f !/ e sin f sin .f !/ v0 (A4) D P p1 e2 C C C C C C K Œcos .f !/ e cos ! v0 D C C C The above expression is a function of time because f is a function of time and it conveniently relates the orbital velocity of a planet to some fundamental systemic parameters.

B. Derivation of the Rossiter-McLaughlin Effect

B.1. The Geometry of a Transiting Extrasolar Planet

Before delving into the derivation of the functional form for the RM Effect, an overview of the transit geometry is helpful. The following is a brief summary of the equations important for the approximation of the RM effect. A more detailed approach for the derivations can be found in Transits and Occultations (Winn 2010). For simplicity, I will assume the planet has no eccentricity. In Figure B.1, a side view of a transit is depicted. With some geometry, the vertical distance at mid-transit from the center of both bodies, or the impact parameter, can be calculated as b a cos i . In Figure B.2 I take the perspective D  2 2 2 of the observer and I define half of the transit chord as l Rp R b . Finally, to D C  approximate the duration of the transit, I assume a constantq Keplerian velocity
trajectory – 75 –

Fig. B.1.— The Impact Parameter of a Transit In the figure above, we have a side view of the transit. The vertical distance from thecenter of the star to the center of the planet is the impact parameter, b. a is the semi-major axis and i is the angle of inclination. subtending an angle ˛. Figure B.3 shows that a triangle is formed with the center of the star, the ingress, and the egress points. Using the figure I derive the duration to be approximately: D P D Tdur  V D 2 a P ˛ (B1)  2 P 1 l sin   a  

B.2. Parameters for the system

Let us consider a set of Cartesian coordinates where the Y -axis is parallel to the spin vector of the star, which is contained in the YZ-plane Figure B.4. The Z-axis is in C the direction of the observer and the X-axis is perpendicular to the spin vector. In such coordinate system, the velocity of the planet is simply a function of X. I normalized the new coordinate system such that the surface of our star, modeled as a sphere, is 1. That is to say in our coordinates: X 2 Y 2 Z2 1. In these coordinates, the vectors for the angular C C D – 76 –

Fig. B.2.— Observational Perspective of a Transiting Exoplanet Above is the observational perspective of the transit. Half of the transit chord length, l, can be derived with the impact parameter and the radii of the two objects.

Fig. B.3.— Geometry of a Transit The length of the transiting arc of an exoplanet is approximated by the chord between points A and B. 2l is the full chord length while ˛ is the angle corresponding to the arc. The term D used in Equation B1 is the arc length between points A and B. The time can be approximated using basic properties of chords and arcs. – 77 –

Fig. B.4.— Alignment Between Planetary and Stellar Spin Vectors In any planetary system, the planet and the star have their spin vectors separated by a certain angle. In reality, this three-dimensional angle, in blue and denoted , depends on the inclination of the star, the inclination of the planet, and the obliquity. A more detailed description of the true angle and it’s true value can be seen in (Fabrycky & Winn 2009). The Z-axis points towards the observer while the X and Y axes are the plane of the sky. velocity of the star and the radius become:

 0;  i ;  i E sin cos  D h    i (B2) R R X;Y;Z E D  h i Above I multiplied the unit vector of position by the radius to obtain a physical quantity. The velocity at any point on the surface of the star is the vector product between radius and angular velocity such that:

v  R E D E   E  R 0; sin i ; cos i X;Y;Z (B3) D   h  i  h i  R Z sin i Y cos i ;X cos i ; X sin i D   h    i  R is simply the equatorial velocity of our star. Given the conformation of our system,   the only radial component apparent to the observer is in the Z-direction. Therefore, on any – 78 – spot of the star we, the observer, measure:

vobs vk DE O  R X sin i (B4) D    V sin i X D   I define l as half the length of the transit chord, b as the impact parameter,  as the displacement from the midpoint of the transit, and  as the angle between the stellar spin

vector and the planetary spin vector (Figure 1.3). There are also temporal parameters Tc, the time of mid-transit, and Tdur , the transit duration time. The system has the following expressions: b a cos i D  2 2 2 l Rp R b D C  qR
Tc  D vorb (B5)

P 1 l Tdur sin D  a   2l .t Tc/  .t/ D Tdur With all of this, one can finally write an expression for the dimensionless X and Y coordinates of our system which are seen in Figure B.5:  .t/ cos  b sin  X .t; / D R  (B6)  .t/ sin  b cos  Y .t; / C D R 

B.3. Analytical Approximation

I defined the coordinate system conveniently such that the radial velocity of the staris only dependent on X .t; /. Now I can solve for the expected radial velocity difference by – 79 –

Fig. B.5.— The Normalized Stellar Coordinate System during a Transit The above figure depicts the coordinate system, normalized to the stellar radius. Thespatial position in terms of X and Y (indicated as the blue points on the left figure) depend on the obliquity, , the displacement from mid-transit along the transit chord, , and the impact parameter, b. As shown on the right, Equation B6 can be derived from the figure. Since  is a function of time, the formulae in Equation B6 are valid for any point on the transit chord. – 80 – assuming the ansatz:

vRM KRM g .t; / (B7) D

I have introduced KRM as the amplitude of the radial velocity given as:

KRM V sin i .ı/ where D   (B8) ı Fractional loss of light D The function g .t; / is a dimensionless function where if I neglect the ingress, egress, and limb darkening: X .t; / in transit when g .t; / 8 (B9) D ˆ <ˆ 0 out of transit Consolidating Equations B8 and B9,ˆ we see that Equation B7 becomes: :ˆ

vRM V sin i X .t; / ı    2 ı , ignoring limb darkening (B10) D 1 2 R p D R  In reality, the expression as shown in Equation B10 is not an ideal approximation because limb darkening is significant in most systems. The limb darkening law can be generalized, with some amount of error, to the following (Ohta et al. 2005; Gimenez 2006):

N n I./ I0 1 un .1  / D ˇ n 1 ˇ ˇ XD ˇ ˇ ˇ (B11)  cosˇ ˇ D ˇ ˇ 1 .X 2 Y 2/ D C In the above expression, is the anglep of incidence, that is the angle between the line of

sight and the normal to the local stellar surface while un are the limb-darkening coefficients. My first attempt at correcting this was to assume a linear limb darkening law suchthat:

I ./ I0 Œ1 u1 .1 / (B12)  – 81 –

In 2005, Ohta et al. developed an expression for the value of the Rossiter-McLaughlin effect assuming a limb darkening law. The expressions I used from the paper are as follows. When the planet was completely in transit I used:

2 Œ1 u1.1 W2/ V sin i 2 1 2   1 u1Œ 3 .1 W1/ 8 v RM ˆW1 0 (B13)  ˆ  <ˆ W2  ˆ  ˆ ˆ For either the ingress or egress periods: of the planet, I used:

u .1 u / y  2 1  1 W 1 Œ 0 cos . / 1 p 4 C C C V sin i 1 1 2 1    .1 3 u1/ .1 u1/Œsin y0 .1 p/y0 cos . / u1W3 8 C C ˆ ˆW3 0 ˆ  ˆ ˆ  g.xc ;p; / ˆW4 2 . / xc g.1 ; ; / W2 .1 / ˆ  ˆ ˆ 2 ˆp 1  ˆ D ˆ 2 2 ˆ 2p p ˆ C C ˆ 2.1 p/ ˆ D C vRM ˆ (B14)  ˆ  ˆxc x0 < D C 2 2 2 p ˆx0 1 ˆ D 2.1 p/ ˆ C ˆ 2 ˆy0 1 x ˆ D 0 ˆ 1 ˆ 2 2 ˆ p 2 1 .X 1 p/ ˆg x;  ; 1 X sin ˆ p 1 X 2 ˆ D ˆ   ˆ
2
2 2 2 2 ˆ X 1 p 1 X X 1 p ˆ C C ˆ r ˆ h
i h
i I also considered:ˆ a model following a quadratic limb darkening model such that:

2 I ./ I0 1 u1 .1 / u2 .1 / (B15)    I made use of an IDL routine from Mandel & Agol which incorporated the quadratic limb darkening law in a calculation for the fractional loss of light. I used this value for ı such – 82 – that:

vRM V sin i X .t; / ı .t; X; Y; ; u1; u2/ (B16)    A plot of the comparison between the three different limb darkening approximations can be seen in Figure B.6. For all of the calculations in this paper, I used the parameterized form of the Rossiter-McLaughlin effect given incorporating the quadratic limb darkening law as shown in Equation B16.

Fig. B.6.— The Limb Darkening Laws For this paper I tried adopting different limb darkening laws. The data is from KELT-7b and the curves are plotted with the measured value of  derived in this paper. The bottom half of the graph shows the different transit night residuals. Ignoring limb darkening results inan overestimate of the amplitude, with half of the residuals lying below zero and the other half above. The linear limb darkening law resulted in an increasing linear trend for the residuals. The quadratic limb darkening law provided the best fit and made the residuals hover around zero. – 83 –

B.4. Addition Offset Velocity for HAT-P-49b & KELT-7b

The functional form I used for both HAT-P-49b and KELT-7b was the same as in Equation B16 with the addition of a correction term. I evaluated the fit with a constant correction to the RM velocity. As shown in this thesis, the radial velocity points recorded for both HAT-P-49b and KELT-7b had a substantial amount of jitter and variability. The addition of a correctional offset velocity became apparent once analysis of the data forHAT- P-49b began. The variation in the measurements suggest that for HAT-P-49b there is more than instrumental jitter in the system. I assumed a correction term due the potential for photospheric jitter, much like in the case of the exoplanetary system HD 189733 (Winn et al. 2006). If the stellar surface has inhomogeneities, such as flows, and has a timescale similar to the period, the residuals in the radial velocity will have a lot of scatter. In that system, Winn et al. assumed a correction term,  , to account for the scatter in the residuals taken on different nights. I assumed that points obtained on the same night have less scatter than those obtained on different nights. For example, if the night-to-night jitter results from rotating star spots, this would produce noise on the timescale of the rotational period. The rotational period is much larger than the duration of transit and I would expect a systemic trend in the points obtained on the same night. The constant correction term was only added to the transit night as it was the only night significant for the RM model.

B.5. The semi-amplitude velocity as a free parameter for the RM Effect in HAT-P-49b & KELT-7b

The functional form I used for both systems was the same as in Equation B16 with the addition of a constant velocity correction. The points outside of transit had a lot of scatter which, like for the case of HAT-P-49b, seemed to imply an active stellar surface. As Lovis & Fischer describe, there may be many different models for stellar noise which include – 84 – p-mode oscillations, granulations, and magnetic oscillations among other variations. I did not completely understand the intrinsic radial velocity noise, there was no model that could account for variations and combining data that was captured over sporadic periods of time will cause scatter. To fix the semi-amplitude velocity, K, for a system with stellar activity is to assume a constant velocity gradient that is always present when observing the star. The inherent noise on the stellar surface has the potential to introduce trends in the radial velocity during the observation if the timescales are roughly a few hours (Albrecht et al. 2012). Albrecht et al. were well aware of the potential to introduce bias by assuming the K derived from the orbital solution was appropriate for the transit night data. K was kept as a free parameter even in the planetary systems where noise was not significant. While having K as a free parameter increases the uncertainties and produces wider confidence intervals for λ, it ensures bias from the stellar activity is limited in the model. – 85 –

C. Correlation Plots

C.1. HAT-P-49b

1.0 = 90.5 14.9 [deg]

0.8

0.6

0.4

0.2 Probability Distribution Function

= 225.7 15.0 [m/s] [m/s]

0.2 0.4 0.6 0.8 1.0 [degrees] Probability Distribution Function

Fig. C.1.— HAT-P-49b Correlation between λ and Δγ The description is the same as for Figure 3.3. There is some correlation between the constant velocity offset and the obliquity for HAT-P-49b.

C.2. KELT-7b

D. Markov Chain Monte Carlo Procedure for Uncertainties

The Markov Chain Monte Carlo (MCMC) algorithm was used to solve for the parameters and their relative uncertainties. Much of the information on MCMC was determined from Ford’s paper (Ford 2005). As Ford describes, MCMC seeks to generate a chain, in this case sequences of parameters, which are sampled from the desired probability distribution. MCMC can recreate the posterior distributions for a model and provide a sense of the uncertainties within each parameter. This is optimal for situations where we know very – 86 –

1.0 = 90.5 14.9 [deg]

0.8

0.6

0.4

0.2 Probability Distribution Function 300 57.5 [m/s]r= 0.019 K= 62.2 57.5 [m/s]r= 200

100

K [m/s] 0

0 50 100 150 200 0.2 0.4 0.6 0.8 1.0 [degrees] Probability Distribution Function

Fig. C.2.— HAT-P-49b Correlation between λ and K.

Fig. C.3.— KELT-7b Correlation between λ and Δγ The description is the same as for Figure 3.7. There is some correlation between the constant velocity offset and the obliquity for KELT-7b. – 87 –

Fig. C.4.— KELT-7b Correlation between λ and K. little about the nature of the error in the data. I utilized the Metropolis-Hastings (MH) algorithm to generate a trial state for the MCMC procedure.

The basic implementation of the MCMC procedure follows Ford’s description of the algorithm in addition to notes from Astronomy 193: Noise and Data Analysis in Astrophysics. I initialized the chain with the parameters derived from χ2-fitting. For HAT- P-49b I had three free parameters: the obliquity, , the stellar rotational velocity, V sin i ,   and the offset velocity, Vı . For KELT-7b there were four free parameters: , Vı , V sin i ,   and the semi-amplitude velocity, K. With the initial state input, the procedure added links to the chain by adding random jumps in each parameter. The jump scale for the procedure was set such that jumping by the scale altered the χ2 value by 1. The actual magnitude  of the jump was determined by multiplying a pseudorandom number taken from a normal distribution by the jump scale. The jumps were accepted if they reduced the value of χ2. I assumed Gaussian distributions for my parameters and that the likelihood of each step

 2 2 2 was given as e , where  is the difference between the test χ and the previous L D χ2. If the jump did not decrease the value of χ2, it was accepted if < μ where μ was a L – 88 – pseudorandom number between 0 and 1.

While I gave an initial arbitrary jump scale, it was set to modify itself such that the acceptance ratio for MCMC was 25%. The recommended acceptance rate for MCMC is  somewhere between 20%-40% (Geyer & Thompson 1995; Gilks et al. 1996). If the acceptance rate is too high, the chain may not mix well and will slowly explore parameter space. If the acceptance rate is too low, the rejection of many candidates results in slow convergence. Convergence to a parameter state occurs in MCMC regardless of the initial state. One must account for the time it takes to converge, which varies depending on the starting point. This is known as burn-in and, for my MCMC procedure, I rejected the first 20% of each chain. Each MCMC procedure was allowed to have a chain of nine million states. With regards to the bounds for MCMC, I only constrained the obliquity for KELT-7b such that    . 2   2 The model I used for this paper (Figure 1.3), assumed that a positive value of  we when the angle increased counterclockwise. This is simply to ensure prograde exoplanets as having positive  and retrograde exoplanets as having negative . For HAT-P-49b I integrated from Œ0; 2. The symmetry of the problem required that I fixed the bounds on angle, or else multiple  exist where χ2 reaches a minimum. – 89 –

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